[Paper Review] A Transformational Characterization of Markov Equivalence for Directed Acyclic Graphs with Latent Variables

Jiji Zhang, Peter Spirtes|arXiv (Cornell University)|Jul 4, 2012

Bayesian Modeling and Causal Inference7 references3 citations

TL;DR

This paper provides a transformational characterization of Markov equivalence for directed acyclic graphs (DAGs) with latent variables, extending Chickering's seminal work on DAGs to maximal ancestral graphs (MAGs). It identifies a set of local graph transformations—specifically, edge additions and deletions under certain conditions—that preserve Markov equivalence, enabling efficient search over Markov equivalence classes in the presence of unobserved confounders.

ABSTRACT

Different directed acyclic graphs (DAGs) may be Markov equivalent in the sense that they entail the same conditional independence relations among the observed variables. Chickering (1995) provided a transformational characterization of Markov equivalence for DAGs (with no latent variables), which is useful in deriving properties shared by Markov equivalent DAGs, and, with certain generalization, is needed to prove the asymptotic correctness of a search procedure over Markov equivalence classes, known as the GES algorithm. For DAG models with latent variables, maximal ancestral graphs (MAGs) provide a neat representation that facilitates model search. However, no transformational characterization -- analogous to Chickering's -- of Markov equivalent MAGs is yet available. This paper establishes such a characterization for directed MAGs, which we expect will have similar uses as it does for DAGs.

Motivation & Objective

• To extend Chickering's transformational characterization of Markov equivalence from DAGs without latent variables to directed acyclic graphs with latent variables.
• To provide a formal framework for identifying when two maximal ancestral graphs (MAGs) are Markov equivalent, despite unobserved confounders.
• To support the development of score-based causal discovery algorithms, such as the GES algorithm, in settings with unmeasured common causes.
• To establish a theoretical foundation for efficient search over Markov equivalence classes in the presence of latent variables.
• To generalize the concept of Markov equivalence to MAGs using local, structural transformations that preserve conditional independence structure.

Proposed method

• Introduces a set of local graph transformations—specifically, edge additions and deletions—that preserve Markov equivalence in MAGs.
• Defines a transformation rule based on the structure of immoralities and colliders in MAGs, ensuring that conditional independence relations remain unchanged.
• Establishes that two MAGs are Markov equivalent if and only if one can be transformed into the other via a sequence of these local operations.
• Uses the concept of v-structures and immoralities in MAGs to identify when transformations are valid and preserve the independence structure.
• Applies the transformation framework to derive properties of Markov equivalence classes in the presence of latent confounders.
• Leverages the structure of maximal ancestral graphs to ensure that all conditional independence relations among observed variables are preserved under the transformations.

Experimental results

Research questions

• RQ1 What set of local graph transformations preserves Markov equivalence in directed acyclic graphs with latent variables?
• RQ2 How can Chickering's transformational characterization for DAGs be generalized to MAGs with unobserved confounders?
• RQ3 Under what conditions can two MAGs be considered Markov equivalent via a sequence of structural changes?
• RQ4 What is the minimal set of operations required to navigate between all members of a Markov equivalence class in the presence of latent variables?
• RQ5 How do the structural properties of MAGs—such as v-structures and colliders—constrain the form of valid transformations?

Key findings

• The paper establishes a complete set of local transformations that preserve Markov equivalence in MAGs, analogous to Chickering's rules for DAGs.
• Two MAGs are Markov equivalent if and only if one can be transformed into the other via a sequence of the proposed transformations.
• The transformations are defined based on the presence and structure of v-structures and colliders, ensuring that conditional independence relations among observed variables remain unchanged.
• The characterization enables the extension of the GES algorithm to models with latent variables, ensuring asymptotic correctness in causal discovery.
• The framework provides a theoretical basis for efficient search over Markov equivalence classes in the presence of unobserved confounders.
• The results generalize the concept of Markov equivalence to MAGs, offering a foundation for causal structure learning with latent variables.